Mathematical Formulation of Motion and Deformation and Its Applications

This chapter is intended to give a summary of Lie groups and Lie algebras for computer graphics, including an example from interpolations and blending of motions and deformation. In animation and filmmaking procedure, we want to get a smooth transition of the drawing of given starting and ending point (the drawings at these points are called key frame). This problem can be understood as an interpolation, so that various approach have been proposed and used in Computer Graphics. After the seminal work so-called ARAP (as-rigid-as-possible), serious properties of matrix groups have been taken into account both in theoretical and in computational point of view. To understand and develop these properties, we here employ a Lie theoretic approach.